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Defensive alliances in graphs: a survey
A set of vertices of a graph is a defensive -alliance in if
every vertex of has at least more neighbors inside of than outside.
This is primarily an expository article surveying the principal known results
on defensive alliances in graph. Its seven sections are: Introduction,
Computational complexity and realizability, Defensive -alliance number,
Boundary defensive -alliances, Defensive alliances in Cartesian product
graphs, Partitioning a graph into defensive -alliances, and Defensive
-alliance free sets.Comment: 25 page
On defensive alliances and line graphs
Let be a simple graph of size and degree sequence . Let denotes the line graph of
. The aim of this paper is to study mathematical properties of the
alliance number, , and the global alliance number,
, of the line graph of a simple graph. We show
that In particular, if is a -regular
graph (), then , and if is a
-semiregular bipartite graph, then . As a consequence of
the study we compare and , and we
characterize the graphs having . Moreover, we show that
the global-connected alliance number of is bounded by
where
denotes the diameter of , and we show that the global
alliance number of is bounded by . The case of
strong alliances is studied by analogy
Partitioning A Graph In Alliances And Its Application To Data Clustering
Any reasonably large group of individuals, families, states, and parties exhibits the phenomenon of subgroup formations within the group such that the members of each group have a strong connection or bonding between each other. The reasons of the formation of these subgroups that we call alliances differ in different situations, such as, kinship and friendship (in the case of individuals), common economic interests (for both individuals and states), common political interests, and geographical proximity. This structure of alliances is not only prevalent in social networks, but it is also an important characteristic of similarity networks of natural and unnatural objects. (A similarity network defines the links between two objects based on their similarities). Discovery of such structure in a data set is called clustering or unsupervised learning and the ability to do it automatically is desirable for many applications in the areas of pattern recognition, computer vision, artificial intelligence, behavioral and social sciences, life sciences, earth sciences, medicine, and information theory. In this dissertation, we study a graph theoretical model of alliances where an alliance of the vertices of a graph is a set of vertices in the graph, such that every vertex in the set is adjacent to equal or more vertices inside the set than the vertices outside it. We study the problem of partitioning a graph into alliances and identify classes of graphs that have such a partition. We present results on the relationship between the existence of such a partition and other well known graph parameters, such as connectivity, subgraph structure, and degrees of vertices. We also present results on the computational complexity of finding such a partition. An alliance cover set is a set of vertices in a graph that contains at least one vertex from every alliance of the graph. The complement of an alliance cover set is an alliance free set, that is, a set that does not contain any alliance as a subset. We study the properties of these sets and present tight bounds on their cardinalities. In addition, we also characterize the graphs that can be partitioned into alliance free and alliance cover sets. Finally, we present an approximate algorithm to discover alliances in a given graph. At each step, the algorithm finds a partition of the vertices into two alliances such that the alliances are strongest among all such partitions. The strength of an alliance is defined as a real number p, such that every vertex in the alliance has at least p times more neighbors in the set than its total number of neighbors in the graph). We evaluate the performance of the proposed algorithm on standard data sets
Defensive alliance polynomial
We introduce a new bivariate polynomial which we call the defensive alliance
polynomial and denote it by da(G; x, y). It is a generalization of the alliance
polynomial [Carballosa et al., 2014] and the strong alliance polynomial
[Carballosa et al., 2016]. We show the relation between da(G; x, y) and the
alliance, the strong alliance and the induced connected subgraph [Tittmann et
al., 2011] polynomials. Then, we investigate information encoded in da(G; x, y)
about G. We discuss the defensive alliance polynomial for the path graphs, the
cycle graphs, the star graphs, the double star graphs, the complete graphs, the
complete bipartite graphs, the regular graphs, the wheel graphs, the open wheel
graphs, the friendship graphs, the triangular book graphs and the quadrilateral
book graphs. Also, we prove that the above classes of graphs are characterized
by its defensive alliance polynomial. A relation between induced subgraphs with
order three and both subgraphs with order three and size three and two
respectively, is proved to characterize the complete bipartite graphs. Finally,
we present the defensive alliance polynomial of the graph formed by attaching a
vertex to a complete graph. We show two pairs of graphs which are not
characterized by the alliance polynomial but characterized by the defensive
alliance polynomial
Alliance Partitions in Graphs.
For a graph G=(V,E), a nonempty subset S contained in V is called a defensive alliance if for each v in S, there are at least as many vertices from the closed neighborhood of v in S as in V-S. If there are strictly more vertices from the closed neighborhood of v in S as in V-S, then S is a strong defensive alliance. A (strong) defensive alliance is called global if it is also a dominating set of G. The alliance partition number (respectively, strong alliance partition number) is the maximum cardinality of a partition of V into defensive alliances (respectively, strong defensive alliances). The global (strong) alliance partition number is defined similarly. For each parameter we give both general bounds and exact values. Our major results include exact values for the alliance partition number of grid graphs and for the global alliance partition number of caterpillars
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